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SUMMARY:Normal numbers and fractal measures - Pablo Shmerkin (Surrey)
DTSTART:20130206T160000Z
DTEND:20130206T170000Z
UID:TALK41711@talks.cam.ac.uk
CONTACT:Bateman
DESCRIPTION:It is known from E. Borel that almost all real numbers are nor
 mal to all integer bases. On the other hand\, it is conjectured that natur
 al constants such as $\\pi$\, $e$ or $\\sqrt{2}$ are normal\, but this pro
 blem is so far untractable. In the talk I will describe a new\ndynamical a
 pproach to an intermediate problem: are ``natural'' fractal measures suppo
 rted on numbers normal to a given base? Our results are formulated in term
 s of an auxiliary flow that reflects the structure of the measure as one z
 ooms in towards a point.\n\nUnlike classical methods based on the Fourier 
 transform\, our approach allows to establish normality in some non-integer
  bases and is robust under smooth perturbations of the measure. As applica
 tions\, we\ncomplete and extend results of B. Host and E. Lindenstrauss on
  normality of $\\times p$ invariant measures\, and many other classical no
 rmality results. This is a joint work with M. Hochman.\n
LOCATION:MR11\, CMS
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